The “5-layer decomposition” in the experiment is implemented via the Mallat algorithm, which decomposes the original signal (sampling frequency 2000 Hz) into 5 detail layers (D1 to D5) and 1 approximation layer. Each detail layer corresponds to a specific frequency band: 1.

D1: 500–1,000 Hz

This division isolates transient fault components (concentrated in 62.5–250 Hz, i.e., D3–D4) from the 50 Hz fundamental wave and noise.

The “Adaptive Threshold” in the experimental section is rooted in the statistical characteristics of wavelet coefficients. For normal signals, detail coefficients d j follow a Gaussian distribution N 0 , σ 2 ; for fault signals, coefficients at mutation points significantly exceed the noise level.

The adaptive threshold for layer j is calculated using Equation 3 (Donoho and Johnstone, 1994): λ j = σ j 2 ⁡ ln ⁡ N (3) where σ j (noise standard deviation) is estimated by σ j = median ( | d j ) / 0.6745 , and N is the signal length. Coefficients exceeding λ j are identified as fault mutation points, directly supporting the fault time localization in the experiment.

For decomposition layers, alternatives like 4-layer miss low-frequency transient components, while 6-layer introduces redundant noise; 5-layer is selected as it balances fault band coverage (62.5–250 Hz) and efficiency. For the detection method, fixed thresholds lack noise adaptability and peak-to-peak ratios are insensitive to minor faults, so adaptive thresholding is chosen for its dynamic adjustment to noise and signal amplitude, ensuring robust detection.

The wavelet transform achieves precise analysis of signals through a flexible scale adjustment mechanism, with its key advantage lying in the adaptive windowing property of wavelet basis functions. This analysis window automatically adjusts its size according to changes in the scale parameter. By adjusting the scale factor, the original signal is decomposed into a series of components with distinct time and frequency characteristics.

Unlike the Fourier transform (global frequency analysis), wavelet transform provides joint time-frequency analysis, making it ideal for capturing transient changes in non-stationary signals—critical for fault detection.

Haar Wavelet: The earliest wavelet function used in signal processing, resembling a step shape. With an extremely short support length, Haar wavelet enables fast computation and is widely applied to signal analysis in practical engineering. Despite its simple structure, it meets the basic requirements of wavelet analysis.

Among Daubechies (dbN) wavelets, db4 is superior for voltage transformer fault detection due to its balanced performance in three key aspects (Gogula and Edward, 2023; Sundararaman and Jain, 2023):

Vanishing Moment: db4’s 4th-order vanishing moment effectively suppresses high-frequency electromagnetic noise in power systems while retaining transient fault components, outperforming db1 (1st-order, poor noise suppression).

To verify the superiority of the Db4 wavelet in the fault localization of voltage transformers, a systematic comparison of the detection performance among the Db4 wavelet, the Sym4 wavelet (Baroumand et al., 2025), and the traditional FFT-based STFT (Manap et al., 2021) was conducted in this study. As shown in Figure 1, in the scenario of phase jump faults, the optimal fault time localization capability was exhibited by the Db4 wavelet in the D3 detail coefficient layer, with the sharpest and most isolated mutation peaks generated at the fault onset and end moments. Quantitative analysis demonstrated that the localization errors of the Db4 wavelet were 2.1 ms (onset) and 1.8 ms (end), which were slightly lower than those of the Sym4 wavelet (2.3 ms and 2.0 ms) and markedly outperformed the time ambiguity associated with the STFT method. This result experimentally confirms that the Db4 wavelet achieves an optimal balance between support length and vanishing moment, thereby providing empirical evidence for its engineering application in the monitoring of voltage transformers.

Multi-scale analysis further reveals the rationality of selecting the Db4 wavelet. As shown in Figure 2, the Db4 wavelet exhibits the most significant response to phase jump faults in the D3 layer (125–250 Hz), with a peak significance ratio of 15.3 for its detail coefficients—higher than 14.7 of the Sym4 wavelet and 7.8 of STFT.

Compared with the Sym4 wavelet, the Db4 wavelet not only maintains comparable detection performance but also exhibits lower theoretical complexity (with a complexity index of 1.8 vs. 2.1 for Sym4), a characteristic that makes it more compatible with the real-time processing requirements of online monitoring systems.

Experimental results further verify that the Db4 wavelet achieves an optimal balance among three core performance metrics: fault feature extraction accuracy, noise robustness, and computational efficiency. This balanced performance renders it an ideal technical choice for fault localization of voltage transformers.

The excellent time-frequency localization performance of the Db4 wavelet originates from its 4th-order vanishing moment. Specifically, this 4th-order vanishing moment enables the wavelet to effectively suppress high-frequency interference noise while simultaneously preserving the integrity of fault transient components—two key functions that jointly lay a reliable foundation for accurate voltage transformer fault localization.

As the core detail layer in the 5-level db4 wavelet decomposition, the D3 layer (125–250 Hz) enhances VT fault measurement accuracy through two key designs: precise frequency band matching with fault features, and optimal time-frequency resolution balance.

The core energy of transient features (e.g., dielectric loss anomalies, capacitor breakdown) of typical VT faults concentrates in 62.5–250 Hz, and the D3 layer exactly covers the high-frequency core components of this range-these are the most direct fault representations (e.g., phase jumps). Meanwhile, the D3 layer sits in the frequency gap between high-frequency noise (D1/D2, 250–1,000 Hz) and fundamental/low-order harmonics (D4/D5, 31.25–125 Hz), isolating interference via wavelet filtering. Based on experimental data, the D3 layer’s fault feature SNR reaches 28.6%–37% higher than D2 (21.0) and 52% higher than D4 (18.8)-confirming its ability to retain fault energy while suppressing interference.

For temporal localization, the D3 layer’s analysis window (scale a = 4) matches the 10–100 ms fault transient duration, balancing time/frequency resolution (avoiding false mutations in D1/D2 or blurred points in D4/D5). For phase jump faults (preset start time: 0.1 s), the D3 layer’s localization error is only 2.1 ms-41% lower than D2 (3.6 ms) and 34% lower than D4 (3.2 ms). Its isolated, prominent fault peaks (unmasked by noise/energy dispersion) also simplify fault timing judgment.

In short, the D3 layer boosts precision via feature separation and temporal localization, providing key support for subsequent fault detection experiments.

The 5-layer decomposition (Gavgani, 2024; Thango, 2025), implemented via the Mallat algorithm, is tailored to db4 and voltage transformer fault characteristics:

Principle: It iteratively decomposes the 2000 Hz sampled signal into low-frequency approximation coefficients and high-frequency detail coefficients, isolating transient fault components from the 50 Hz fundamental wave and noise.

The experimental data are obtained from voltage transformer detection devices collected by a power generation company. The parameter settings for wavelet transform are shown in Table 1. The experimental design is grounded in the wavelet theory outlined in Section 2. The five decomposition layers correspond to analyzing the signal at five different scales. As the level j increases (larger scale a), the frequency resolution improves, allowing the capture of slower signal variations, while the time resolution decreases. The translation parameter b, inherently defined by the sampling points, enables the precise time localization of fault-induced mutations. The “Adaptive Thresholding” detection method is employed based on the statistical properties of the wavelet coefficients (e.g., their energy or magnitude distribution), which change significantly at the onset and termination of a fault.

The specific experimental parameter settings for the dielectric loss fault are shown in Table 2. The sampling frequency of 2000 Hz is chosen to satisfy the Nyquist sampling theorem for the 50 Hz fundamental power frequency and its significant harmonics, which is a standard configuration in power system monitoring to ensure signal fidelity.

Daubechies 4 wavelet is selected to perform 5-layer decomposition on the voltage signal in Figure 3. The normal voltage and dielectric loss fault voltage are shown in Figure 3.

In Figure 3, due to the subtle nature of the phase shift (+5°) caused by dielectric loss faults, it is difficult to directly identify the onset and end times from the original voltage waveform. However, wavelet transform effectively captures these minor transient changes. The normal voltage signal and the dielectric loss fault voltage signal are subjected to Daubechies 4 wavelet transform to obtain Figure 4.

As shown in Figure 4, comparing the detail coefficients (D1-D5) reveals that the third layer (D3, 125–250 Hz) shows the most significant response to the phase shift characteristic of dielectric loss. This is because the insulation deterioration caused by dielectric loss anomalies excites high-frequency oscillation components concentrated in the 100–250 Hz band, which is precisely covered by the D3 layer. In contrast, D1 and D2 layers (>250 Hz) mainly contain electromagnetic interference noise, while D4 and D5 layers (<125 Hz) include power frequency drifts, making fault features less distinguishable. More importantly, the D3 layer demonstrates excellent signal-to-noise separation capability, with a peak significance ratio (defined as the ratio of peak mutation value to background coefficient RMS) reaching up to 15.3, significantly higher than Sym4 wavelets (14.7) and STFT (7.8) (see Figure 2). Therefore, the sharp peaks in the D3 plot for the fault signal, absent in the normal signal, clearly mark the fault boundaries at 0.1s and 0.2s, achieving a precise fault time localization error of only 2.1 ms (onset) and 1.8 ms (end).

The experimental conditions for the capacitor breakdown fault are shown in Table 3. The voltage drop to 55 V represents a significant deviation from the normal 80V, simulating a severe fault condition commonly defined in engineering standards.

The normal voltage and capacitor breakdown fault voltage are shown in Figure 5.

The fault occurs at 0.1s, with the voltage dropping significantly to 55V, a value indicative of a critical failure, and lasts until 0.2s. The Daubechies 4 wavelet transform of the normal voltage signal and capacitor breakdown voltage signal is shown in Figure 6.

As shown in Figure 6, comparing the detail coefficients reveals significant differences in the 4th layer (D4), which is sensitive to the frequency components altered by the sudden voltage sag. The peak mutation in D4 confirms the fault ends at 0.2s. For even clearer identification of the fault start time, the difference between the detail coefficients of the normal and fault signals is plotted in Figure 7.

The peak mutations in the detail coefficient difference plot of Figure 7 accurately determine the start and end times of the capacitor breakdown fault, demonstrating the enhanced sensitivity of this approach.

The experimental parameters for the short-circuit fault of the compensation reactor are shown in Table 4. The increase in voltage to 100 V with a phase lag of 5° simulates another common fault scenario in voltage transformers.

The normal voltage and compensation reactor short-circuit fault voltage are shown in Figure 8.

The fault occurs at 0.1s, with the voltage amplitude increasing to 100 V and phase lagging by 5°, conditions representative of a reactor fault, until 0.2s. The Daubechies 4 wavelet transform results are shown in Figure 9.

As shown in Figure 9, the 3rd layer (D3) detail coefficients show the most significant difference, effectively capturing the combined effect of overvoltage and phase shift. Figure 10 reveals two distinct peaks in the detail coefficient difference plots of layers 3 and 4, corresponding to the start and end times of the fault.

In summary, across three typical VT faults, the third detail layer (D3, 125–250 Hz) consistently isolates fault transient characteristics and achieves high-precision time localization. Its universality stems from its alignment with resonant/oscillatory energy excited by VT faults, combined with the good smoothness and temporal localization capabilities of the db4 wavelet at this scale. Therefore, D3 can be considered a general sensitive frequency band for VT fault diagnosis, providing optimal input for adaptive threshold detection.